Heat oil in a large nonstick skillet over medium-high heat. Add onion, and cook for 5 minutes or until golden. Add mushrooms and salt; cook 5 minutes, stirring constantly. Add vinegar; remove mixture from pan. Set aside.
Combine the paprika and the next 4 ingredients (paprika through black pepper). Divide the ground round into 4 equal portions, shaping each into a 1/2-inch-thick patty. Coat patties with spice mixture. Grill patties 4 minutes on each side or until done. Place burgers on muffin halves, and top each burger with 1/4 cup onion mixture.